Research Article | Open Access
Strong Uniform Attractors for Nonautonomous Suspension Bridge-Type Equations
We discuss long-term dynamical behavior of the solutions for the nonautonomous suspension bridge-type equation in the strong Hilbert space , where the nonlinearity is translation compact and the time-dependent external forces only satisfy condition () instead of translation compact. The existence of strong solutions and strong uniform attractors is investigated using a new process scheme. Since the solutions of the nonautonomous suspension bridge-type equation have no higher regularity and the process associated with the solutions is not continuous in the strong Hilbert space, the results are new and appear to be optimal.
Consider the following equations: Suspension bridge equations (1.1) have been posed as a new problem in the field of nonlinear analysis  by Lazer and McKenna in 1990. This model has been derived as follows. In the suspension bridge system, suspension bridge can be considered as an elastic and unloaded beam with hinged ends. denotes the deflection in the downward direction; represents the viscous damping. The restoring force can be modeled owing to the cable with one-sided Hooke’s law so that it strongly resists expansion but does not resist compression. The simplest function to model the restoring force of the stays in the suspension bridge can be denoted by a constant times , the expansion, if is positive, but zero, if is negative, corresponding to compression; that is, , where Besides, the right-hand side of (1.1) also contains two terms: the large positive term corresponding to gravity, and a small oscillatory forcing term possibly aerodynamic in origin, where is small.
In the study of equations of mathematical physics, attractor is a proper mathematical concept about the depiction of the behavior of the solutions of these equations when time is large or tends to infinity, which describes all the possible limits of solutions. In the past two decades, many authors have proved the existence of attractor and discussed its properties for various mathematical physics models (e.g., see [10–12] and the reference therein). About the long-time behavior of suspension bridge-type equations, for the autonomous case, in [13, 14] the authors have discussed long-time behavior of the solutions of the problem on and obtained the existence of global attractors in the space and .
It is well known that, for a model to describe the real world which is affected by many kinds of factors, the corresponding nonautonomous model is more natural and precise than the autonomous one, moreover, it always presents as a nonlinear equation, not just a linear one. Therefore, in this paper, we will discuss the following nonautonomous suspension bridge-type equation: let be an open bounded subset of with smooth boundary, , and we add the nonlinear forcing term (which is dependent on deflection and time ) to (1.1) and neglect gravity, then we can obtain the following initial-boundary value problem: where is an unknown function, which could represent the deflection of the road bed in the vertical plane; and are time dependant external forces; represents the restoring force, denotes the spring constant; represents the viscous damping, is a given positive constant.
To our knowledge, this is the first time to consider the nonautonomous dynamics of (1.3) with the time dependant external forces and in the strong topological space . At the same time, in mathematics, we only assume that the force term satisfies the so-called condition (introduced in ), which is weaker than translation compact assumption (see  or Section 2 below).
This paper is organized as follows. At first, in Section 2, we give (recall) some preliminaries, including the notation we will use, the assumption on nonlinearity , and some general abstract results about nonautonomous dynamical system. Then, in Section 3 we prove our main result about the existence of strong attractor for the nonautonomous dynamical system generated by the solution of (1.3).
2. Notation and Preliminaries
With the usual notation, we introduce the spaces , , , where . We equip these spaces with inner product and norm , and , respectively, Obviously, we have where is dual space of , respectively, the injections are continuous, and each space is dense in the following one.
In the following, the assumption on the nonlinearity is given. Let be a function from to and satisfy where , and there exists , such that
Suppose that is an arbitrary positive constant, and where is a sufficiently small constant.
By virtue of (2.5), we can get
When , problem (1.3) is equivalent to the following equations in :
From the Poincaré inequality, there exists a proper constant , such that
We introduce the Hilbert space and endow this space with norm:
Let be a Banach space, and let a two-parameter family of mappings on :
Definition 2.1 (see ). Let be a parameter set. , is said to be a family of processes in Banach space , if for each , is a process; that is, the two-parameter family of mappings from to satisfy where is called the symbol space and is the symbol.
Note that the following translation identity is valid for a general family of processes , , if a problem is the unique solvability and for the translation semigroup satisfying :
A set is said to be a uniformly (w.r.t ) absorbing set for the family of processes , if for any and , there exists such that for all . A set is said to be uniformly (w.r.t. ) attracting for the family of processes , if for any fixed and every ,
Definition 2.2 (see ). A closed set is said to be the uniform (w.r.t. ) attractor of the family of processes , if it is uniformly (w.r.t. ) attracting (attracting property) and contained in any closed uniformly (w.r.t. ) attracting set of the family of processes , : (minimality property).
Now we recalled the results in .
Definition 2.3 (see [16, 17]). A family of processes , is said to be satisfying uniform (w.r.t. ). Condition (C) if for any fixed , and , there exist a and a finite dimensional subspace of such that (i) is bounded;(ii), where and is abounded projector.
Theorem 2.4 (see ). Let be a complete metric space, and let be a continuous invariant semigroup on satisfying the translation identity. A family of processes , possesses compact uniform (w.r.t. ) attractor in satisfying if it (i)has a bounded uniformly (w.r.t. ) absorbing set ;(ii)satisfies uniform (w.r.t. ) condition (C), where . Moreover, if is a uniformly convex Banach space, then the converse is true.
Let be a Banach space. Consider the space of functions with values in that are 2-power integrable in the Bochner sense. is a set of all translation compact functions in , is a set of all translation bound functions in .
In , the authors have introduced a new class of functions which are translation bounded but not translation compact. In the third section, let the forcing term satisfy condition , we can prove the existence of compact uniform (w.r.t. , ) attractor for nonautonomous suspension bridge equation in .
Definition 2.5 (see ). Let be a Banach space. A function is said to satisfy condition () if for any , there exists a finite dimensional subspace of such that where is the canonical projector.
Denote by the set of all functions satisfying condition (). From , we can see that .
Remark 2.6. In fact, the function satisfying condition () implies the dissipative property in some sense, and the condition () is very natural in view of the compact condition, uniform condition ().
Lemma 2.7 (see ). If , then for any and we have where is the canonical projector and is a positive constant.
In order to define the family of processes of (2.10), we also need the following results.
Proposition 2.8 (see ). If is reflexive separable, then (i) for all , ;(ii) the translation group is weakly continuous on ;(iii) for all .
Proposition 2.9 (see ). Let , then (i) for all , , and the set is bound in ;(ii) the translation group is continuous on with the topology of ;(iii) for all .
3. Uniform Attractors in
To describe the asymptotic behavior of the solutions of our system, we set and , where denotes the closure of a set in topological space . If , then , that is to be where denotes the norm in .
3.1. Existence and Uniqueness of Strong Solutions
At first, we give the concept of strong solutions for the initial-boundary value problem (2.10).
Definition 3.1. Set , for . We suppose that , , satisfying (2.3)–(2.6) and . The function is said to be a strong solution to problem (2.10) in the time interval , with initial data , provided for all and a.e. .
Then, by using the methods in  (Galerkin approximation method), we can get the following result about the existence and uniqueness of strong solutions.
Theorem 3.2 (existence and uniqueness of strong solutions). Define , for all . Let , , satisfying (2.3)–(2.6). Then for any given , there is a unique solution for problem (2.10) in . Furthermore, for , let and be two initial conditions, and denote by corresponding solutions to problem (2.10). Then the estimates hold as follows: for all ,
Thus, (2.10) will be written as an evolutionary system introduced and for brevity, as , the system (2.10) can be written in the operator form where is the symbol of (3.4). If , then problem (3.4) has a unique solution . This implies that the process given by the formula is defined in .
Now we define the symbol space. A fixed symbol can be given, where is in , the function satisfying (2.3)–(2.6), and is a Banach space, endowed with the following norm: Obviously, the function is in . we define , where denotes the closure of a set in topological space (or ). So, if , then and all satisfy condition .
Applying Propositions 2.8 and 2.9 and Theorem 3.2, we can easily know that the family of processes , , are defined. Furthermore, the translation semigroup satisfies that for all , , and the following translation identity: holds.
Then for any , the problem (3.4) with instead of possesses a corresponding to process acting on .
3.2. A Priori Estimates
3.2.1. A Priori Estimates in
Theorem 3.3. Assume that is a solution of (2.10) with initial data . If the nonlinearity satisfies (2.3)–(2.6), , , , then there is a positive constant such that for any bounded (in ) subset , there exists such that
Proof. Now we will prove that are bounded in .
We assume that is positive and satisfies Multiplying (2.10) by and integrating over , we have
We can easily see that Then, substituting (3.12)-(3.13) into (3.11), we can obtain that In view of (2.6) and (2.8), we can know
We introduce the functional as follows:
Setting , we choose proper positive constants and , such that hold, then .
We define , then where , .
Analogous to the proof of Lemma 2.1.3 in , we can estimate the integral and obtain where .
By virtue of (2.7), we can get Choosing , we obtain from (3.17) In consideration of (2.9) and , we can see
Combining (3.20), (3.22), and (3.24), we can deduce that
Assume that , as , we have
We complete the proof.
3.2.2. A Priori Estimates in
Lemma 3.4. Assuming that is a strong solution of (2.10) with initial data . If the nonlinearity satisfies (2.3)–(2.6), , , , then there is a positive constant such that for any bounded (in ) subset , there exists such that
Proof. Now we will prove that are bounded in .
We assume that is positive and satisfies Multiplying (2.10) by and integrating over , we have where .
We can deduce that Then, substituting (3.30) into (3.29), we have In view of (2.5) and Theorem 3.3, we can see that Exploiting and Theorem 3.3, we can obtain Consequently, Choose small enough such that This leads to By (2.5), (2.9), the Hölder inequality, and Theorem 3.3, we have
We introduce the functional as follows:
Setting , we define , then where .
Analogous to the proof of Theorem 3.3, we can estimate the integral and obtain where .
Assuming that , as , we have Applying (2.9), the Hölder inequality, the Cauchy inequality, and Theorem 3.3, we can deduce from (3.41) that where depends on , , , , , and .
We complete the proof.
Theorem 3.5 (bounded uniformly absorbing set in ). Presuming that and . Let satisfy (2.3)–(2.6), , and , be the family of processes corresponding to (2.10) in , then has a uniformly (w.r.t. ) absorbing set in . That is, for any bounded subset , there exists such that
3.3. The Existence of Uniform Attractor
We will show the existence of uniform attractor to problem (2.10) in .
Theorem 3.6 (uniform attractor). Let be the family of processes corresponding to problem (2.10). If satisfyies (2.3)–(2.6), , and , then possesses a compact uniform (w.r.t. ) attractor in , which attracts any bounded set in with , satisfying where is the uniformly (w.r.t. ) absorbing set in .
Proof. From Theorems 2.4 and 3.5, we merely need to prove that the family of processes , satisfies uniform (w.r.t. ) condition () in . We assume that , are eigenvalue of operator in , satisfying
denotes eigenvector corresponding to eigenvalue , , which forms an orthogonal basis in ; at the same time they are also a group of canonical basis in or , and satisfy
Let , is an orthogonal projector. For any , we write where .
Choose , and . Taking the scalar product with for (2.10) in , we have where
Clearly, we can get that
Combining (3.49)–(3.52), we obtain from (3.48)